"Practice" in this sense just means "(traditional/conventional) behavior".

Beyond that, this sounds like a homework question, which we won't answer here. You should be able to do this yourself.

Several questions in one. Let's see:

First, "Topic-Prominent languages" doesn't precisely categorize a class of languages, in part because in general languages don't fall precisely into classes (e.g. "tonal languages"), and because especially for something as complex in usage and communication like information structure, it varies across languages. (The reasons for not having a precise categorization are borderline cases, internal variation, and a lack of corresponding other features in the languages that independently vary.)

Second, yes, there are some languages that use topicalization more than others, in various ways. They probably develop differently in some ways too. One relatively familiar language is probably French, which has a structure generally similar to other Romance languages (even fairly similar to English), but much more often uses topicalization than most other European languages. So literal equivalents of "Me, I like that", etc., are common. Why? That's much harder to answer, but there's been a lot written about French so it's a reasonable place to start. One factor may be that French has lot a lot of the complex morphology of Latin, most of which is still retained in the other Romance languages, so now because most verb forms don't indicate person, etc., subject pronouns are obligatory (like English, unlike Spanish or Italian), and topicalization is another way to reinforce the sentence structure. But French really is at the low end of the extent of 'topic-prominent' languages.

At the extreme, consider for example Arabic. In the present tense, the linking verb "be" is omitted. In fact, it's more than that: Arabic grammar is deeply rooted in topicalization, such that very often the verb "be" is not needed. So literally you can say "I student", meaning "I'm a student". Arabic isn't alone (compare Russian for example), but it is an extreme case. For one thing, when you do add "be", such as in the past tense, the grammar of the sentence actually changes-- for example, the word "student" in "I was (a) student" would be marked in the accusative case, but in the present ("I student") it would be nominative. That suggests those really are fundamentally different structures, not just a "silent" verb "be" in the present tense. Furthermore, Arabic allows topics even if they aren't the same as the subject. Something like "I, my teacher (is) good". Traditional Arab grammarians noticed that there seem to be two types of sentences in Arabic: nominal sentences (beginning with a noun and corresponding to a topic-comment structure) and verbal sentences (beginning with a verb and corresponding to a topic-comment structure). A typical verbal sentence is VSO (verb-subject-object) order. Another common order is SVO, but I personally consider this (in line with the traditional analysis) to be a combination of topic-comment plus V(S)O (without an overt subject in the comment part, as is normal because of verb agreement). In summary, for Arabic this tells us two things: (1) the omission of "be" can potentially lead to more use of topic structures, and (2) those structures might become integrated with the rest of the grammar. (There's also a cycle, such as in Modern Hebrew a trend to use a pronoun like "he" as a sort of replacement for "be" in those sentences, something like "John he student" meaning "John IS a student".)

Case-marking is another complicated issue, which basically means how subjects, objects and other parts of the sentence are marked as belonging to those roles, typically by suffixes. (Look that up for background, but also just compare English "I" vs. "me" or "who" vs. "whom", which is a sort of contrast much more widely used in other languages.) There is a lot of variation around the world, including for example ergative languages (very complicated topic if you're not familiar with it, look it up if you want). But then there are also languages that behave quite differently from what we'd expect from any typical case configuration, and some of these appear to revolve around topics rather than subjects at all. So we can ask "what is a subject anyway?" Well, a few languages don't really seem to care. They use topics instead. These "focus" systems (or various other terms) are especially common in Austronesian languages for example. (That's a large and widespread language family spanning from Polynesia to Madagascar, originating in Taiwan historically, unrelated to Chinese.) Common examples often come from languages of the Philippines, so you might want to look up something about those systems. The result is roughly that there are semantic roles in a sentence like "agent", "patient", etc., but not clearly grammatical roles like "subject" or "object"; instead, every sentence seems to have something like "passivization" going on, but for any of the various semantic roles, such that the language is topic-prominent: whichever entity is considered important functions something like a "subject", and becomes the focused element or topic of the sentence, with the rest appearing in a normal configuration. Again, compare passivization in English like "The lamp was broken (by me)", where the focus is shifted to the lamp rather than the person who broke it: that sort of transformation is available in these languages but for any (or at least many) of the various entities involved, not just the specific case of English passives.

Overall, how does all of this develop? Well, topics are generally fronted elements, joined prosodically (by intonation) or sometimes a special grammatical linking element (called a "topic marker" or similar). Note also that sometimes a similar category of "focus" is distinguished, although the two often overlap in some ways and I would assume one can easily develop into the other as languages change. The result then is that this is a way to add or highlight information regardless of whether it is easy to express within a simple sentence. So when there is a need to express information beyond what is easily expressed in a simple sentence, topics are available. And eventually, in various ways, that can become systematic. The factors are different for different languages, but that's probably a reasonable overview for most languages.

So, could this be a real historical factor in language change?

I can't comment about this specific issue, although as I suggested above, it is likely that topics are used to add or clarify information when it isn't clear from a simple sentence, so something along those lines could be a factor.

But more generally, there have been some popular theories recently about how language learning, especially by adults who do not ever gain native-like competence in a language, can eventually shape a language to look very different. This has been phrased as a "loss of complexity" for globalized languages like English, although that must be interpreted in a specific way. So what you're suggesting could be a factor, although it's more often presented as a case of learners changing the language through usage, rather than speakers changing it to facilitate the understanding of learners, but I suppose those factors are inter-related. For more about that, see among others:
Trudgill, Peter. 2011. Sociolinguistic Typology: Social Determinants of Linguistic Complexity. Oxford: Oxford University Press.
McWhorter, John H. 2007. Language Interrupted: Signs of non-native acquisition in standard language grammars. New York: Oxford University Press.
Note that those books are written in a reasonably accessible way for a general but motivated audience, not just for specialists.
The theories promoted there are not yet generally accepted among linguists, but they are considered to be relevant factors for understanding these complex phenomena.

Setting aside debuccalization in [bʌʔə], prenasal debuccalization as in [ɹaʔn̩] is the overwhelming majority rule in the US.

I was referring to the more general phenomenon, which is more common in British dialects. Consonant clusters do work differently sometimes, but I believe also fall into the more general pattern where applicable. That would also be less perceptually salient, although you're right to emphasize that the two examples in the original post involve nasals, so perhaps that's more specifically what is going on here. (In that case, I'd emphasize even more that this doesn't matter, because it's less salient, and much more common.)

But while we're on the topic, while it is true that there is no correlation between accent and intelligence, there is a correlation between dialect choice and professionalism. There is the same correlation between speech pattern choices and professionalism as there is between professionalism and other choices such as dress, cleanliness posture and snarkiness. Perhaps you disagree.

That's a problematic and complicated argument. Yes, some people do "posture" by changing their accent to appeal to others in professional settings, and yes, there are some widespread "expectations" i professional settings. But that does NOT mean that individuals with a different accent are less intelligent or necessarily less professional. Rather, it is a systemic problem, in part due to demographics, and in part due to perpetuating these beliefs so that it's generally seen as "OK" to judge someone by their accent. Something similar can be said about race, given statistical disparity in wealth distribution, etc. But from that you cannot say that an individual employee of a certain race is "less professional" or "less educated" or whatever. Statistics are misleading in cases like these, because they perpetuate stereotypes and confuse effect with cause. There is absolutely no reason that someone with a particular accent would not be an exemplary employee.

(As you say, the OP's social context is important to consider any further details, because another complication is the distinction between different kinds of identities expressed via accents, such as geographic vs. social class vs. peer group, etc.)

I've assumed it's to annoy me (seriously....we're very close and love poking fun at each other).

Then it's not a serious problem.

Drove me up the wall and made me question her professionalism. 

To be blunt, that's your problem, not hers.
It can be difficult or uncomfortable to get over linguistic prejudices, but it's important to think about. There's simply no relationship between accents and intelligence. It's similar to another myth, that people who don't speak your language (or don't speak it well) are unintelligent-- in fact, if anything they probably speak more languages than you do (their language, as well as you speak your own, plus at least some of your language, and maybe others). Perception can be deceiving, especially if you don't take the time to question it.

I'm mildly shocked that Daniel associates this process with Britain.

Huh? It's a stereotypical feature of some British accents (though not all!), especially Cockney (the examples of "butter" as 'buh'er" or "water" as "wah-er" are well known by most English speakers, I'd think, and certainly linguists). It's relatively rare (but not unattested) in the US.

Your response = LIKE A BOSS

JK.  I've noticed she doesn't do it with her friends much because they don't speak that way.  But when she's around me, she does.  I've assumed it's to annoy me (seriously....we're very close and love poking fun at each other).  But I want to make sure she understands that adults in professional situations (she's about to go to university) don't speak this way.  I went to an osteo surgeon before a knee surgery once.  His young nurse tech came in and was asking med questions about "Sta'ins" (meaning statins) and the like.  Drove me up the wall and made me question her professionalism. 

Is there a professional term for this similar to VocalFry, Upspeak/talk, Valtalk, etc?